Polytope of Type {3,2,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,9,6}*648
if this polytope has a name.
Group : SmallGroup(648,554)
Rank : 5
Schlafli Type : {3,2,9,6}
Number of vertices, edges, etc : 3, 3, 9, 27, 6
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,9,6,2} of size 1296
   {3,2,9,6,3} of size 1944
Vertex Figure Of :
   {2,3,2,9,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,9,2}*216, {3,2,3,6}*216
   9-fold quotients : {3,2,3,2}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,18,6}*1296b, {6,2,9,6}*1296
   3-fold covers : {3,2,9,18}*1944, {9,2,9,6}*1944, {3,6,9,6}*1944, {3,2,9,6}*1944a, {3,2,27,6}*1944
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7, 8)( 9,12)(10,14)(11,13)(15,18)(16,20)(17,19)(21,24)(22,26)
(23,25)(27,30)(28,29);;
s3 := ( 4,10)( 5, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)(18,27)(20,23)
(21,25)(24,29)(26,28);;
s4 := ( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(27,28)(29,30);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(30)!(2,3);
s1 := Sym(30)!(1,2);
s2 := Sym(30)!( 5, 6)( 7, 8)( 9,12)(10,14)(11,13)(15,18)(16,20)(17,19)(21,24)
(22,26)(23,25)(27,30)(28,29);
s3 := Sym(30)!( 4,10)( 5, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)(18,27)
(20,23)(21,25)(24,29)(26,28);
s4 := Sym(30)!( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(27,28)(29,30);
poly := sub<Sym(30)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope